LMFDB - Embedded newform 592.4.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,4,Mod(1,592)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 592 = 2^{4} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	592.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$34.9291307234$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 172x^{6} - 325x^{5} + 7345x^{4} + 20706x^{3} - 73170x^{2} - 269101x - 182948 $ x^8 - 172x^6 - 325x^5 + 7345x^4 + 20706x^3 - 73170x^2 - 269101x - 182948
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.00248$ of defining polynomial
Character	$\chi$	$=$	592.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.270938 q^{3} -20.1562 q^{5} -12.1826 q^{7} -26.9266 q^{9} +O(q^{10})$	$q-0.270938 q^{3} -20.1562 q^{5} -12.1826 q^{7} -26.9266 q^{9} -13.6829 q^{11} +31.4637 q^{13} +5.46108 q^{15} -74.5832 q^{17} -108.342 q^{19} +3.30072 q^{21} -83.8400 q^{23} +281.273 q^{25} +14.6108 q^{27} +30.9993 q^{29} -214.509 q^{31} +3.70722 q^{33} +245.554 q^{35} -37.0000 q^{37} -8.52471 q^{39} +514.883 q^{41} -83.8023 q^{43} +542.738 q^{45} +505.926 q^{47} -194.585 q^{49} +20.2074 q^{51} +406.739 q^{53} +275.796 q^{55} +29.3540 q^{57} -79.5428 q^{59} +88.9920 q^{61} +328.035 q^{63} -634.189 q^{65} -1073.52 q^{67} +22.7154 q^{69} +920.355 q^{71} -133.528 q^{73} -76.2074 q^{75} +166.693 q^{77} +45.1424 q^{79} +723.059 q^{81} -747.964 q^{83} +1503.31 q^{85} -8.39888 q^{87} +727.198 q^{89} -383.309 q^{91} +58.1186 q^{93} +2183.77 q^{95} +812.977 q^{97} +368.435 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9}+O(q^{10}) $ 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9	$ 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9} - 33 q^{11} + 50 q^{13} - 22 q^{15} + 78 q^{17} - 158 q^{19} + 373 q^{21} - 76 q^{23} + 678 q^{25} - 469 q^{27} + 358 q^{29} - 390 q^{31} + 333 q^{33} - 546 q^{35} - 296 q^{37} - 442 q^{39} + 1141 q^{41} + 4 q^{43} + 1770 q^{45} + 735 q^{47} + 731 q^{49} - 132 q^{51} + 1349 q^{53} + 882 q^{55} + 2648 q^{57} - 1146 q^{59} + 1920 q^{61} + 534 q^{63} + 1748 q^{65} - 184 q^{67} + 1024 q^{69} + 881 q^{71} + 985 q^{73} + 1803 q^{75} + 1903 q^{77} + 658 q^{79} + 2296 q^{81} + 2773 q^{83} - 592 q^{85} + 3730 q^{87} + 234 q^{89} + 638 q^{91} - 904 q^{93} + 3660 q^{95} + 2174 q^{97} + 3582 q^{99}+O(q^{100}) $ 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9 - 33 * q^11 + 50 * q^13 - 22 * q^15 + 78 * q^17 - 158 * q^19 + 373 * q^21 - 76 * q^23 + 678 * q^25 - 469 * q^27 + 358 * q^29 - 390 * q^31 + 333 * q^33 - 546 * q^35 - 296 * q^37 - 442 * q^39 + 1141 * q^41 + 4 * q^43 + 1770 * q^45 + 735 * q^47 + 731 * q^49 - 132 * q^51 + 1349 * q^53 + 882 * q^55 + 2648 * q^57 - 1146 * q^59 + 1920 * q^61 + 534 * q^63 + 1748 * q^65 - 184 * q^67 + 1024 * q^69 + 881 * q^71 + 985 * q^73 + 1803 * q^75 + 1903 * q^77 + 658 * q^79 + 2296 * q^81 + 2773 * q^83 - 592 * q^85 + 3730 * q^87 + 234 * q^89 + 638 * q^91 - 904 * q^93 + 3660 * q^95 + 2174 * q^97 + 3582 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.270938	−0.0521420	−0.0260710	−	0.999660i	$-0.508300\pi$
$3$	−0.270938	−0.0521420	−0.0260710	+	0.999660i	$0.508300\pi$
$4$	0	0
$5$	−20.1562	−1.80283	−0.901413	−	0.432960i	$-0.857469\pi$
$5$	−20.1562	−1.80283	−0.901413	+	0.432960i	$0.857469\pi$
$6$	0	0
$7$	−12.1826	−0.657797	−0.328899	−	0.944365i	$-0.606677\pi$
$7$	−12.1826	−0.657797	−0.328899	+	0.944365i	$0.606677\pi$
$8$	0	0
$9$	−26.9266	−0.997281
$10$	0	0
$11$	−13.6829	−0.375051	−0.187525	−	0.982260i	$-0.560047\pi$
$11$	−13.6829	−0.375051	−0.187525	+	0.982260i	$0.560047\pi$
$12$	0	0
$13$	31.4637	0.671266	0.335633	−	0.941993i	$-0.391050\pi$
$13$	31.4637	0.671266	0.335633	+	0.941993i	$0.391050\pi$
$14$	0	0
$15$	5.46108	0.0940030
$16$	0	0
$17$	−74.5832	−1.06406	−0.532032	−	0.846724i	$-0.678571\pi$
$17$	−74.5832	−1.06406	−0.532032	+	0.846724i	$0.678571\pi$
$18$	0	0
$19$	−108.342	−1.30818	−0.654090	−	0.756417i	$-0.726948\pi$
$19$	−108.342	−1.30818	−0.654090	+	0.756417i	$0.726948\pi$
$20$	0	0
$21$	3.30072	0.0342989
$22$	0	0
$23$	−83.8400	−0.760081	−0.380040	−	0.924970i	$-0.624090\pi$
$23$	−83.8400	−0.760081	−0.380040	+	0.924970i	$0.624090\pi$
$24$	0	0
$25$	281.273	2.25018
$26$	0	0
$27$	14.6108	0.104142
$28$	0	0
$29$	30.9993	0.198497	0.0992487	−	0.995063i	$-0.468356\pi$
$29$	30.9993	0.198497	0.0992487	+	0.995063i	$0.468356\pi$
$30$	0	0
$31$	−214.509	−1.24281	−0.621403	−	0.783491i	$-0.713437\pi$
$31$	−214.509	−1.24281	−0.621403	+	0.783491i	$0.713437\pi$
$32$	0	0
$33$	3.70722	0.0195559
$34$	0	0
$35$	245.554	1.18589
$36$	0	0
$37$	−37.0000	−0.164399
$37$	−37.0000	−0.164399
$38$	0	0
$39$	−8.52471	−0.0350012
$40$	0	0
$41$	514.883	1.96125	0.980625	−	0.195894i	$-0.0627608\pi$
$41$	514.883	1.96125	0.980625	+	0.195894i	$0.0627608\pi$
$42$	0	0
$43$	−83.8023	−0.297203	−0.148602	−	0.988897i	$-0.547477\pi$
$43$	−83.8023	−0.297203	−0.148602	+	0.988897i	$0.547477\pi$
$44$	0	0
$45$	542.738	1.79792
$46$	0	0
$47$	505.926	1.57015	0.785073	−	0.619403i	$-0.212625\pi$
$47$	505.926	1.57015	0.785073	+	0.619403i	$0.212625\pi$
$48$	0	0
$49$	−194.585	−0.567303
$50$	0	0
$51$	20.2074	0.0554824
$52$	0	0
$53$	406.739	1.05415	0.527075	−	0.849819i	$-0.323289\pi$
$53$	406.739	1.05415	0.527075	+	0.849819i	$0.323289\pi$
$54$	0	0
$55$	275.796	0.676151
$56$	0	0
$57$	29.3540	0.0682112
$58$	0	0
$59$	−79.5428	−0.175518	−0.0877592	−	0.996142i	$-0.527971\pi$
$59$	−79.5428	−0.175518	−0.0877592	+	0.996142i	$0.527971\pi$
$60$	0	0
$61$	88.9920	0.186791	0.0933956	−	0.995629i	$-0.470228\pi$
$61$	88.9920	0.186791	0.0933956	+	0.995629i	$0.470228\pi$
$62$	0	0
$63$	328.035	0.656009
$64$	0	0
$65$	−634.189	−1.21018
$66$	0	0
$67$	−1073.52	−1.95749	−0.978743	−	0.205091i	$-0.934251\pi$
$67$	−1073.52	−1.95749	−0.978743	+	0.205091i	$0.934251\pi$
$68$	0	0
$69$	22.7154	0.0396322
$70$	0	0
$71$	920.355	1.53839	0.769197	−	0.639012i	$-0.220656\pi$
$71$	920.355	1.53839	0.769197	+	0.639012i	$0.220656\pi$
$72$	0	0
$73$	−133.528	−0.214087	−0.107043	−	0.994254i	$-0.534138\pi$
$73$	−133.528	−0.214087	−0.107043	+	0.994254i	$0.534138\pi$
$74$	0	0
$75$	−76.2074	−0.117329
$76$	0	0
$77$	166.693	0.246707
$78$	0	0
$79$	45.1424	0.0642900	0.0321450	−	0.999483i	$-0.489766\pi$
$79$	45.1424	0.0642900	0.0321450	+	0.999483i	$0.489766\pi$
$80$	0	0
$81$	723.059	0.991851
$82$	0	0
$83$	−747.964	−0.989154	−0.494577	−	0.869134i	$-0.664677\pi$
$83$	−747.964	−0.989154	−0.494577	+	0.869134i	$0.664677\pi$
$84$	0	0
$85$	1503.31	1.91832
$86$	0	0
$87$	−8.39888	−0.0103501
$88$	0	0
$89$	727.198	0.866099	0.433049	−	0.901370i	$-0.357438\pi$
$89$	727.198	0.866099	0.433049	+	0.901370i	$0.357438\pi$
$90$	0	0
$91$	−383.309	−0.441557
$92$	0	0
$93$	58.1186	0.0648024
$94$	0	0
$95$	2183.77	2.35842
$96$	0	0
$97$	812.977	0.850982	0.425491	−	0.904963i	$-0.360101\pi$
$97$	812.977	0.850982	0.425491	+	0.904963i	$0.360101\pi$
$98$	0	0
$99$	368.435	0.374031
$100$	0	0
$101$	−932.299	−0.918487	−0.459243	−	0.888310i	$-0.651879\pi$
$101$	−932.299	−0.918487	−0.459243	+	0.888310i	$0.651879\pi$
$102$	0	0
$103$	−621.994	−0.595018	−0.297509	−	0.954719i	$-0.596156\pi$
$103$	−621.994	−0.595018	−0.297509	+	0.954719i	$0.596156\pi$
$104$	0	0
$105$	−66.5300	−0.0618349
$106$	0	0
$107$	−826.016	−0.746299	−0.373150	−	0.927771i	$-0.621722\pi$
$107$	−826.016	−0.746299	−0.373150	+	0.927771i	$0.621722\pi$
$108$	0	0
$109$	−1234.90	−1.08516	−0.542578	−	0.840006i	$-0.682552\pi$
$109$	−1234.90	−1.08516	−0.542578	+	0.840006i	$0.682552\pi$
$110$	0	0
$111$	10.0247	0.00857210
$112$	0	0
$113$	−855.109	−0.711875	−0.355937	−	0.934510i	$-0.615838\pi$
$113$	−855.109	−0.711875	−0.355937	+	0.934510i	$0.615838\pi$
$114$	0	0
$115$	1689.90	1.37029
$116$	0	0
$117$	−847.211	−0.669441
$118$	0	0
$119$	908.615	0.699938
$120$	0	0
$121$	−1143.78	−0.859337
$122$	0	0
$123$	−139.501	−0.102264
$124$	0	0
$125$	−3149.86	−2.25386
$126$	0	0
$127$	2461.02	1.71953	0.859764	−	0.510692i	$-0.170611\pi$
$127$	2461.02	1.71953	0.859764	+	0.510692i	$0.170611\pi$
$128$	0	0
$129$	22.7052	0.0154968
$130$	0	0
$131$	1274.19	0.849822	0.424911	−	0.905235i	$-0.360305\pi$
$131$	1274.19	0.849822	0.424911	+	0.905235i	$0.360305\pi$
$132$	0	0
$133$	1319.89	0.860517
$134$	0	0
$135$	−294.497	−0.187750
$136$	0	0
$137$	−17.4059	−0.0108546	−0.00542732	−	0.999985i	$-0.501728\pi$
$137$	−17.4059	−0.0108546	−0.00542732	+	0.999985i	$0.501728\pi$
$138$	0	0
$139$	−1049.84	−0.640621	−0.320310	−	0.947313i	$-0.603787\pi$
$139$	−1049.84	−0.640621	−0.320310	+	0.947313i	$0.603787\pi$
$140$	0	0
$141$	−137.075	−0.0818706
$142$	0	0
$143$	−430.516	−0.251759
$144$	0	0
$145$	−624.828	−0.357856
$146$	0	0
$147$	52.7204	0.0295803
$148$	0	0
$149$	2308.58	1.26931	0.634653	−	0.772797i	$-0.281143\pi$
$149$	2308.58	1.26931	0.634653	+	0.772797i	$0.281143\pi$
$150$	0	0
$151$	352.471	0.189958	0.0949791	−	0.995479i	$-0.469722\pi$
$151$	352.471	0.189958	0.0949791	+	0.995479i	$0.469722\pi$
$152$	0	0
$153$	2008.27	1.06117
$154$	0	0
$155$	4323.69	2.24056
$156$	0	0
$157$	−270.520	−0.137515	−0.0687575	−	0.997633i	$-0.521903\pi$
$157$	−270.520	−0.137515	−0.0687575	+	0.997633i	$0.521903\pi$
$158$	0	0
$159$	−110.201	−0.0549655
$160$	0	0
$161$	1021.39	0.499979
$162$	0	0
$163$	−2603.77	−1.25119	−0.625593	−	0.780150i	$-0.715143\pi$
$163$	−2603.77	−1.25119	−0.625593	+	0.780150i	$0.715143\pi$
$164$	0	0
$165$	−74.7236	−0.0352559
$166$	0	0
$167$	992.954	0.460102	0.230051	−	0.973179i	$-0.426111\pi$
$167$	992.954	0.460102	0.230051	+	0.973179i	$0.426111\pi$
$168$	0	0
$169$	−1207.03	−0.549401
$170$	0	0
$171$	2917.29	1.30462
$172$	0	0
$173$	790.436	0.347374	0.173687	−	0.984801i	$-0.444432\pi$
$173$	790.436	0.347374	0.173687	+	0.984801i	$0.444432\pi$
$174$	0	0
$175$	−3426.62	−1.48016
$176$	0	0
$177$	21.5512	0.00915189
$178$	0	0
$179$	−4518.51	−1.88676	−0.943378	−	0.331720i	$-0.892371\pi$
$179$	−4518.51	−1.88676	−0.943378	+	0.331720i	$0.892371\pi$
$180$	0	0
$181$	2176.73	0.893896	0.446948	−	0.894560i	$-0.352511\pi$
$181$	2176.73	0.893896	0.446948	+	0.894560i	$0.352511\pi$
$182$	0	0
$183$	−24.1113	−0.00973967
$184$	0	0
$185$	745.780	0.296383
$186$	0	0
$187$	1020.52	0.399078
$188$	0	0
$189$	−177.997	−0.0685045
$190$	0	0
$191$	1204.99	0.456494	0.228247	−	0.973603i	$-0.426701\pi$
$191$	1204.99	0.456494	0.228247	+	0.973603i	$0.426701\pi$
$192$	0	0
$193$	−580.287	−0.216425	−0.108212	−	0.994128i	$-0.534513\pi$
$193$	−580.287	−0.216425	−0.108212	+	0.994128i	$0.534513\pi$
$194$	0	0
$195$	171.826	0.0631011
$196$	0	0
$197$	3466.75	1.25379	0.626894	−	0.779105i	$-0.284326\pi$
$197$	3466.75	1.25379	0.626894	+	0.779105i	$0.284326\pi$
$198$	0	0
$199$	−1380.50	−0.491764	−0.245882	−	0.969300i	$-0.579078\pi$
$199$	−1380.50	−0.491764	−0.245882	+	0.969300i	$0.579078\pi$
$200$	0	0
$201$	290.858	0.102067
$202$	0	0
$203$	−377.651	−0.130571
$204$	0	0
$205$	−10378.1	−3.53579
$206$	0	0
$207$	2257.53	0.758014
$208$	0	0
$209$	1482.44	0.490634
$210$	0	0
$211$	2481.10	0.809508	0.404754	−	0.914426i	$-0.367357\pi$
$211$	2481.10	0.809508	0.404754	+	0.914426i	$0.367357\pi$
$212$	0	0
$213$	−249.359	−0.0802150
$214$	0	0
$215$	1689.14	0.535805
$216$	0	0
$217$	2613.27	0.817513
$218$	0	0
$219$	36.1779	0.0111629
$220$	0	0
$221$	−2346.66	−0.714270
$222$	0	0
$223$	−861.902	−0.258822	−0.129411	−	0.991591i	$-0.541309\pi$
$223$	−861.902	−0.258822	−0.129411	+	0.991591i	$0.541309\pi$
$224$	0	0
$225$	−7573.72	−2.24406
$226$	0	0
$227$	−5280.54	−1.54397	−0.771986	−	0.635640i	$-0.780736\pi$
$227$	−5280.54	−1.54397	−0.771986	+	0.635640i	$0.780736\pi$
$228$	0	0
$229$	86.8789	0.0250704	0.0125352	−	0.999921i	$-0.496010\pi$
$229$	86.8789	0.0250704	0.0125352	+	0.999921i	$0.496010\pi$
$230$	0	0
$231$	−45.1635	−0.0128638
$232$	0	0
$233$	−2099.00	−0.590173	−0.295087	−	0.955471i	$-0.595348\pi$
$233$	−2099.00	−0.590173	−0.295087	+	0.955471i	$0.595348\pi$
$234$	0	0
$235$	−10197.6	−2.83070
$236$	0	0
$237$	−12.2308	−0.00335221
$238$	0	0
$239$	−6120.22	−1.65642	−0.828210	−	0.560418i	$-0.810640\pi$
$239$	−6120.22	−1.65642	−0.828210	+	0.560418i	$0.810640\pi$
$240$	0	0
$241$	3587.54	0.958896	0.479448	−	0.877570i	$-0.340837\pi$
$241$	3587.54	0.958896	0.479448	+	0.877570i	$0.340837\pi$
$242$	0	0
$243$	−590.395	−0.155859
$244$	0	0
$245$	3922.09	1.02275
$246$	0	0
$247$	−3408.85	−0.878137
$248$	0	0
$249$	202.652	0.0515765
$250$	0	0
$251$	5017.86	1.26185	0.630925	−	0.775844i	$-0.282676\pi$
$251$	5017.86	1.26185	0.630925	+	0.775844i	$0.282676\pi$
$252$	0	0
$253$	1147.18	0.285069
$254$	0	0
$255$	−407.305	−0.100025
$256$	0	0
$257$	−286.091	−0.0694391	−0.0347196	−	0.999397i	$-0.511054\pi$
$257$	−286.091	−0.0694391	−0.0347196	+	0.999397i	$0.511054\pi$
$258$	0	0
$259$	450.755	0.108141
$260$	0	0
$261$	−834.705	−0.197958
$262$	0	0
$263$	−3946.21	−0.925224	−0.462612	−	0.886561i	$-0.653088\pi$
$263$	−3946.21	−0.925224	−0.462612	+	0.886561i	$0.653088\pi$
$264$	0	0
$265$	−8198.32	−1.90045
$266$	0	0
$267$	−197.025	−0.0451601
$268$	0	0
$269$	3019.24	0.684336	0.342168	−	0.939639i	$-0.388839\pi$
$269$	3019.24	0.684336	0.342168	+	0.939639i	$0.388839\pi$
$270$	0	0
$271$	−4013.21	−0.899576	−0.449788	−	0.893135i	$-0.648501\pi$
$271$	−4013.21	−0.899576	−0.449788	+	0.893135i	$0.648501\pi$
$272$	0	0
$273$	103.853	0.0230237
$274$	0	0
$275$	−3848.63	−0.843932
$276$	0	0
$277$	4168.93	0.904284	0.452142	−	0.891946i	$-0.350660\pi$
$277$	4168.93	0.904284	0.452142	+	0.891946i	$0.350660\pi$
$278$	0	0
$279$	5776.00	1.23943
$280$	0	0
$281$	6797.21	1.44302	0.721508	−	0.692406i	$-0.243449\pi$
$281$	6797.21	1.44302	0.721508	+	0.692406i	$0.243449\pi$
$282$	0	0
$283$	−2017.53	−0.423779	−0.211890	−	0.977294i	$-0.567962\pi$
$283$	−2017.53	−0.423779	−0.211890	+	0.977294i	$0.567962\pi$
$284$	0	0
$285$	−591.666	−0.122973
$286$	0	0
$287$	−6272.60	−1.29010
$288$	0	0
$289$	649.652	0.132231
$290$	0	0
$291$	−220.266	−0.0443719
$292$	0	0
$293$	−6208.37	−1.23787	−0.618937	−	0.785441i	$-0.712436\pi$
$293$	−6208.37	−1.23787	−0.618937	+	0.785441i	$0.712436\pi$
$294$	0	0
$295$	1603.28	0.316429
$296$	0	0
$297$	−199.918	−0.0390586
$298$	0	0
$299$	−2637.92	−0.510217
$300$	0	0
$301$	1020.93	0.195499
$302$	0	0
$303$	252.595	0.0478918
$304$	0	0
$305$	−1793.74	−0.336752
$306$	0	0
$307$	828.063	0.153942	0.0769708	−	0.997033i	$-0.475475\pi$
$307$	828.063	0.153942	0.0769708	+	0.997033i	$0.475475\pi$
$308$	0	0
$309$	168.522	0.0310254
$310$	0	0
$311$	−7295.74	−1.33024	−0.665118	−	0.746738i	$-0.731619\pi$
$311$	−7295.74	−1.33024	−0.665118	+	0.746738i	$0.731619\pi$
$312$	0	0
$313$	6497.54	1.17336	0.586682	−	0.809818i	$-0.300434\pi$
$313$	6497.54	1.17336	0.586682	+	0.809818i	$0.300434\pi$
$314$	0	0
$315$	−6611.94	−1.18267
$316$	0	0
$317$	−9539.77	−1.69024	−0.845121	−	0.534575i	$-0.820472\pi$
$317$	−9539.77	−1.69024	−0.845121	+	0.534575i	$0.820472\pi$
$318$	0	0
$319$	−424.161	−0.0744465
$320$	0	0
$321$	223.799	0.0389136
$322$	0	0
$323$	8080.51	1.39199
$324$	0	0
$325$	8849.88	1.51047
$326$	0	0
$327$	334.581	0.0565822
$328$	0	0
$329$	−6163.48	−1.03284
$330$	0	0
$331$	−8702.66	−1.44514	−0.722570	−	0.691298i	$-0.757039\pi$
$331$	−8702.66	−1.44514	−0.722570	+	0.691298i	$0.757039\pi$
$332$	0	0
$333$	996.284	0.163952
$334$	0	0
$335$	21638.1	3.52901
$336$	0	0
$337$	−4233.86	−0.684371	−0.342185	−	0.939632i	$-0.611167\pi$
$337$	−4233.86	−0.684371	−0.342185	+	0.939632i	$0.611167\pi$
$338$	0	0
$339$	231.681	0.0371186
$340$	0	0
$341$	2935.11	0.466115
$342$	0	0
$343$	6549.17	1.03097
$344$	0	0
$345$	−457.857	−0.0714499
$346$	0	0
$347$	5297.41	0.819539	0.409769	−	0.912189i	$-0.365609\pi$
$347$	5297.41	0.819539	0.409769	+	0.912189i	$0.365609\pi$
$348$	0	0
$349$	3762.84	0.577135	0.288568	−	0.957459i	$-0.406821\pi$
$349$	3762.84	0.577135	0.288568	+	0.957459i	$0.406821\pi$
$350$	0	0
$351$	459.709	0.0699072
$352$	0	0
$353$	1764.80	0.266092	0.133046	−	0.991110i	$-0.457524\pi$
$353$	1764.80	0.266092	0.133046	+	0.991110i	$0.457524\pi$
$354$	0	0
$355$	−18550.9	−2.77346
$356$	0	0
$357$	−246.178	−0.0364962
$358$	0	0
$359$	11945.0	1.75607	0.878037	−	0.478592i	$-0.158853\pi$
$359$	11945.0	1.75607	0.878037	+	0.478592i	$0.158853\pi$
$360$	0	0
$361$	4879.05	0.711335
$362$	0	0
$363$	309.893	0.0448076
$364$	0	0
$365$	2691.43	0.385961
$366$	0	0
$367$	13454.7	1.91370	0.956850	−	0.290583i	$-0.0938494\pi$
$367$	13454.7	1.91370	0.956850	+	0.290583i	$0.0938494\pi$
$368$	0	0
$369$	−13864.1	−1.95592
$370$	0	0
$371$	−4955.13	−0.693417
$372$	0	0
$373$	−9131.05	−1.26753	−0.633764	−	0.773526i	$-0.718491\pi$
$373$	−9131.05	−1.26753	−0.633764	+	0.773526i	$0.718491\pi$
$374$	0	0
$375$	853.418	0.117521
$376$	0	0
$377$	975.352	0.133245
$378$	0	0
$379$	−10184.9	−1.38038	−0.690188	−	0.723630i	$-0.742472\pi$
$379$	−10184.9	−1.38038	−0.690188	+	0.723630i	$0.742472\pi$
$380$	0	0
$381$	−666.783	−0.0896597
$382$	0	0
$383$	13197.8	1.76077	0.880387	−	0.474256i	$-0.157283\pi$
$383$	13197.8	1.76077	0.880387	+	0.474256i	$0.157283\pi$
$384$	0	0
$385$	−3359.90	−0.444770
$386$	0	0
$387$	2256.51	0.296395
$388$	0	0
$389$	10277.4	1.33955	0.669777	−	0.742562i	$-0.266390\pi$
$389$	10277.4	1.33955	0.669777	+	0.742562i	$0.266390\pi$
$390$	0	0
$391$	6253.06	0.808774
$392$	0	0
$393$	−345.227	−0.0443114
$394$	0	0
$395$	−909.899	−0.115904
$396$	0	0
$397$	2338.02	0.295571	0.147786	−	0.989019i	$-0.452785\pi$
$397$	2338.02	0.295571	0.147786	+	0.989019i	$0.452785\pi$
$398$	0	0
$399$	−357.608	−0.0448691
$400$	0	0
$401$	−3612.85	−0.449918	−0.224959	−	0.974368i	$-0.572225\pi$
$401$	−3612.85	−0.449918	−0.224959	+	0.974368i	$0.572225\pi$
$402$	0	0
$403$	−6749.25	−0.834253
$404$	0	0
$405$	−14574.1	−1.78813
$406$	0	0
$407$	506.268	0.0616579
$408$	0	0
$409$	10632.8	1.28548	0.642738	−	0.766086i	$-0.277799\pi$
$409$	10632.8	1.28548	0.642738	+	0.766086i	$0.277799\pi$
$410$	0	0
$411$	4.71592	0.000565983 0
$412$	0	0
$413$	969.035	0.115455
$414$	0	0
$415$	15076.1	1.78327
$416$	0	0
$417$	284.441	0.0334033
$418$	0	0
$419$	687.393	0.0801464	0.0400732	−	0.999197i	$-0.487241\pi$
$419$	687.393	0.0801464	0.0400732	+	0.999197i	$0.487241\pi$
$420$	0	0
$421$	−7805.33	−0.903583	−0.451791	−	0.892124i	$-0.649215\pi$
$421$	−7805.33	−0.903583	−0.451791	+	0.892124i	$0.649215\pi$
$422$	0	0
$423$	−13622.9	−1.56588
$424$	0	0
$425$	−20978.2	−2.39434
$426$	0	0
$427$	−1084.15	−0.122871
$428$	0	0
$429$	116.643	0.0131272
$430$	0	0
$431$	−3907.62	−0.436713	−0.218357	−	0.975869i	$-0.570070\pi$
$431$	−3907.62	−0.436713	−0.218357	+	0.975869i	$0.570070\pi$
$432$	0	0
$433$	13820.2	1.53385	0.766925	−	0.641737i	$-0.221786\pi$
$433$	13820.2	1.53385	0.766925	+	0.641737i	$0.221786\pi$
$434$	0	0
$435$	169.290	0.0186593
$436$	0	0
$437$	9083.42	0.994323
$438$	0	0
$439$	−5334.01	−0.579905	−0.289952	−	0.957041i	$-0.593639\pi$
$439$	−5334.01	−0.579905	−0.289952	+	0.957041i	$0.593639\pi$
$440$	0	0
$441$	5239.51	0.565761
$442$	0	0
$443$	14573.2	1.56296	0.781481	−	0.623929i	$-0.214464\pi$
$443$	14573.2	1.56296	0.781481	+	0.623929i	$0.214464\pi$
$444$	0	0
$445$	−14657.5	−1.56143
$446$	0	0
$447$	−625.483	−0.0661842
$448$	0	0
$449$	199.350	0.0209530	0.0104765	−	0.999945i	$-0.496665\pi$
$449$	199.350	0.0209530	0.0104765	+	0.999945i	$0.496665\pi$
$450$	0	0
$451$	−7045.11	−0.735568
$452$	0	0
$453$	−95.4977	−0.00990480
$454$	0	0
$455$	7726.05	0.796051
$456$	0	0
$457$	11533.1	1.18052	0.590260	−	0.807214i	$-0.299025\pi$
$457$	11533.1	1.18052	0.590260	+	0.807214i	$0.299025\pi$
$458$	0	0
$459$	−1089.72	−0.110814
$460$	0	0
$461$	10580.1	1.06890	0.534449	−	0.845201i	$-0.320519\pi$
$461$	10580.1	1.06890	0.534449	+	0.845201i	$0.320519\pi$
$462$	0	0
$463$	12655.2	1.27027	0.635136	−	0.772401i	$-0.280944\pi$
$463$	12655.2	1.27027	0.635136	+	0.772401i	$0.280944\pi$
$464$	0	0
$465$	−1171.45	−0.116827
$466$	0	0
$467$	4459.01	0.441838	0.220919	−	0.975292i	$-0.429094\pi$
$467$	4459.01	0.441838	0.220919	+	0.975292i	$0.429094\pi$
$468$	0	0
$469$	13078.3	1.28763
$470$	0	0
$471$	73.2942	0.00717031
$472$	0	0
$473$	1146.66	0.111466
$474$	0	0
$475$	−30473.7	−2.94364
$476$	0	0
$477$	−10952.1	−1.05128
$478$	0	0
$479$	−6418.04	−0.612208	−0.306104	−	0.951998i	$-0.599026\pi$
$479$	−6418.04	−0.612208	−0.306104	+	0.951998i	$0.599026\pi$
$480$	0	0
$481$	−1164.16	−0.110356
$482$	0	0
$483$	−276.733	−0.0260699
$484$	0	0
$485$	−16386.5	−1.53417
$486$	0	0
$487$	16223.5	1.50956	0.754780	−	0.655978i	$-0.227744\pi$
$487$	16223.5	1.50956	0.754780	+	0.655978i	$0.227744\pi$
$488$	0	0
$489$	705.461	0.0652394
$490$	0	0
$491$	12279.1	1.12861	0.564305	−	0.825567i	$-0.309144\pi$
$491$	12279.1	1.12861	0.564305	+	0.825567i	$0.309144\pi$
$492$	0	0
$493$	−2312.03	−0.211214
$494$	0	0
$495$	−7426.24	−0.674313
$496$	0	0
$497$	−11212.3	−1.01195
$498$	0	0
$499$	15379.6	1.37973	0.689865	−	0.723938i	$-0.257670\pi$
$499$	15379.6	1.37973	0.689865	+	0.723938i	$0.257670\pi$
$500$	0	0
$501$	−269.029	−0.0239907
$502$	0	0
$503$	5608.77	0.497182	0.248591	−	0.968608i	$-0.420032\pi$
$503$	5608.77	0.497182	0.248591	+	0.968608i	$0.420032\pi$
$504$	0	0
$505$	18791.6	1.65587
$506$	0	0
$507$	327.032	0.0286469
$508$	0	0
$509$	−10920.0	−0.950923	−0.475462	−	0.879736i	$-0.657719\pi$
$509$	−10920.0	−0.950923	−0.475462	+	0.879736i	$0.657719\pi$
$510$	0	0
$511$	1626.72	0.140826
$512$	0	0
$513$	−1582.96	−0.136237
$514$	0	0
$515$	12537.0	1.07271
$516$	0	0
$517$	−6922.55	−0.588885
$518$	0	0
$519$	−214.159	−0.0181128
$520$	0	0
$521$	−1022.81	−0.0860078	−0.0430039	−	0.999075i	$-0.513693\pi$
$521$	−1022.81	−0.0860078	−0.0430039	+	0.999075i	$0.513693\pi$
$522$	0	0
$523$	−16987.8	−1.42031	−0.710156	−	0.704044i	$-0.751376\pi$
$523$	−16987.8	−1.42031	−0.710156	+	0.704044i	$0.751376\pi$
$524$	0	0
$525$	928.403	0.0771787
$526$	0	0
$527$	15998.8	1.32242
$528$	0	0
$529$	−5137.85	−0.422277
$530$	0	0
$531$	2141.82	0.175041
$532$	0	0
$533$	16200.1	1.31652
$534$	0	0
$535$	16649.4	1.34545
$536$	0	0
$537$	1224.24	0.0983793
$538$	0	0
$539$	2662.49	0.212767
$540$	0	0
$541$	10523.5	0.836301	0.418150	−	0.908378i	$-0.362678\pi$
$541$	10523.5	0.836301	0.418150	+	0.908378i	$0.362678\pi$
$542$	0	0
$543$	−589.759	−0.0466095
$544$	0	0
$545$	24890.9	1.95635
$546$	0	0
$547$	13968.5	1.09186	0.545930	−	0.837831i	$-0.316176\pi$
$547$	13968.5	1.09186	0.545930	+	0.837831i	$0.316176\pi$
$548$	0	0
$549$	−2396.25	−0.186283
$550$	0	0
$551$	−3358.53	−0.259670
$552$	0	0
$553$	−549.950	−0.0422898
$554$	0	0
$555$	−202.060	−0.0154540
$556$	0	0
$557$	−6821.29	−0.518900	−0.259450	−	0.965757i	$-0.583541\pi$
$557$	−6821.29	−0.518900	−0.259450	+	0.965757i	$0.583541\pi$
$558$	0	0
$559$	−2636.73	−0.199502
$560$	0	0
$561$	−276.497	−0.0208087
$562$	0	0
$563$	−17242.7	−1.29075	−0.645376	−	0.763865i	$-0.723299\pi$
$563$	−17242.7	−1.29075	−0.645376	+	0.763865i	$0.723299\pi$
$564$	0	0
$565$	17235.7	1.28339
$566$	0	0
$567$	−8808.72	−0.652437
$568$	0	0
$569$	−240.411	−0.0177127	−0.00885637	−	0.999961i	$-0.502819\pi$
$569$	−240.411	−0.0177127	−0.00885637	+	0.999961i	$0.502819\pi$
$570$	0	0
$571$	−570.032	−0.0417778	−0.0208889	−	0.999782i	$-0.506650\pi$
$571$	−570.032	−0.0417778	−0.0208889	+	0.999782i	$0.506650\pi$
$572$	0	0
$573$	−326.479	−0.0238025
$574$	0	0
$575$	−23581.9	−1.71032
$576$	0	0
$577$	3450.11	0.248926	0.124463	−	0.992224i	$-0.460279\pi$
$577$	3450.11	0.248926	0.124463	+	0.992224i	$0.460279\pi$
$578$	0	0
$579$	157.222	0.0112848
$580$	0	0
$581$	9112.13	0.650662
$582$	0	0
$583$	−5565.38	−0.395360
$584$	0	0
$585$	17076.6	1.20689
$586$	0	0
$587$	15946.9	1.12129	0.560647	−	0.828055i	$-0.310552\pi$
$587$	15946.9	1.12129	0.560647	+	0.828055i	$0.310552\pi$
$588$	0	0
$589$	23240.4	1.62581
$590$	0	0
$591$	−939.275	−0.0653750
$592$	0	0
$593$	22195.0	1.53700	0.768499	−	0.639851i	$-0.221004\pi$
$593$	22195.0	1.53700	0.768499	+	0.639851i	$0.221004\pi$
$594$	0	0
$595$	−18314.2	−1.26187
$596$	0	0
$597$	374.030	0.0256416
$598$	0	0
$599$	15717.2	1.07210	0.536048	−	0.844188i	$-0.319917\pi$
$599$	15717.2	1.07210	0.536048	+	0.844188i	$0.319917\pi$
$600$	0	0
$601$	−24120.8	−1.63712	−0.818560	−	0.574420i	$-0.805227\pi$
$601$	−24120.8	−1.63712	−0.818560	+	0.574420i	$0.805227\pi$
$602$	0	0
$603$	28906.3	1.95216
$604$	0	0
$605$	23054.2	1.54924
$606$	0	0
$607$	2278.98	0.152390	0.0761951	−	0.997093i	$-0.475723\pi$
$607$	2278.98	0.152390	0.0761951	+	0.997093i	$0.475723\pi$
$608$	0	0
$609$	102.320	0.00680823
$610$	0	0
$611$	15918.3	1.05399
$612$	0	0
$613$	13409.0	0.883500	0.441750	−	0.897138i	$-0.354358\pi$
$613$	13409.0	0.883500	0.441750	+	0.897138i	$0.354358\pi$
$614$	0	0
$615$	2811.82	0.184363
$616$	0	0
$617$	−13765.8	−0.898202	−0.449101	−	0.893481i	$-0.648256\pi$
$617$	−13765.8	−0.898202	−0.449101	+	0.893481i	$0.648256\pi$
$618$	0	0
$619$	12858.9	0.834967	0.417483	−	0.908685i	$-0.362912\pi$
$619$	12858.9	0.834967	0.417483	+	0.908685i	$0.362912\pi$
$620$	0	0
$621$	−1224.97	−0.0791566
$622$	0	0
$623$	−8859.14	−0.569717
$624$	0	0
$625$	28330.2	1.81314
$626$	0	0
$627$	−401.649	−0.0255826
$628$	0	0
$629$	2759.58	0.174931
$630$	0	0
$631$	5106.67	0.322176	0.161088	−	0.986940i	$-0.448500\pi$
$631$	5106.67	0.322176	0.161088	+	0.986940i	$0.448500\pi$
$632$	0	0
$633$	−672.225	−0.0422094
$634$	0	0
$635$	−49604.8	−3.10001
$636$	0	0
$637$	−6122.36	−0.380811
$638$	0	0
$639$	−24782.0	−1.53421
$640$	0	0
$641$	−28338.4	−1.74618	−0.873088	−	0.487562i	$-0.837886\pi$
$641$	−28338.4	−1.74618	−0.873088	+	0.487562i	$0.837886\pi$
$642$	0	0
$643$	7603.64	0.466342	0.233171	−	0.972436i	$-0.425090\pi$
$643$	7603.64	0.466342	0.233171	+	0.972436i	$0.425090\pi$
$644$	0	0
$645$	−457.651	−0.0279380
$646$	0	0
$647$	−25829.1	−1.56947	−0.784736	−	0.619830i	$-0.787201\pi$
$647$	−25829.1	−1.56947	−0.784736	+	0.619830i	$0.787201\pi$
$648$	0	0
$649$	1088.38	0.0658283
$650$	0	0
$651$	−708.034	−0.0426268
$652$	0	0
$653$	−3673.85	−0.220167	−0.110083	−	0.993922i	$-0.535112\pi$
$653$	−3673.85	−0.220167	−0.110083	+	0.993922i	$0.535112\pi$
$654$	0	0
$655$	−25682.9	−1.53208
$656$	0	0
$657$	3595.47	0.213505
$658$	0	0
$659$	−1398.47	−0.0826659	−0.0413329	−	0.999145i	$-0.513160\pi$
$659$	−1398.47	−0.0826659	−0.0413329	+	0.999145i	$0.513160\pi$
$660$	0	0
$661$	21484.7	1.26424	0.632118	−	0.774872i	$-0.282186\pi$
$661$	21484.7	1.26424	0.632118	+	0.774872i	$0.282186\pi$
$662$	0	0
$663$	635.800	0.0372435
$664$	0	0
$665$	−26603.9	−1.55136
$666$	0	0
$667$	−2598.98	−0.150874
$668$	0	0
$669$	233.522	0.0134955
$670$	0	0
$671$	−1217.67	−0.0700561
$672$	0	0
$673$	24216.6	1.38705	0.693523	−	0.720435i	$-0.256058\pi$
$673$	24216.6	1.38705	0.693523	+	0.720435i	$0.256058\pi$
$674$	0	0
$675$	4109.61	0.234339
$676$	0	0
$677$	−1421.76	−0.0807132	−0.0403566	−	0.999185i	$-0.512849\pi$
$677$	−1421.76	−0.0807132	−0.0403566	+	0.999185i	$0.512849\pi$
$678$	0	0
$679$	−9904.15	−0.559774
$680$	0	0
$681$	1430.70	0.0805058
$682$	0	0
$683$	9852.43	0.551966	0.275983	−	0.961163i	$-0.410997\pi$
$683$	9852.43	0.551966	0.275983	+	0.961163i	$0.410997\pi$
$684$	0	0
$685$	350.837	0.0195690
$686$	0	0
$687$	−23.5388	−0.00130722
$688$	0	0
$689$	12797.5	0.707615
$690$	0	0
$691$	21509.6	1.18417	0.592085	−	0.805875i	$-0.298305\pi$
$691$	21509.6	1.18417	0.592085	+	0.805875i	$0.298305\pi$
$692$	0	0
$693$	−4488.48	−0.246036
$694$	0	0
$695$	21160.8	1.15493
$696$	0	0
$697$	−38401.6	−2.08689
$698$	0	0
$699$	568.700	0.0307728
$700$	0	0
$701$	−10917.5	−0.588227	−0.294114	−	0.955770i	$-0.595024\pi$
$701$	−10917.5	−0.588227	−0.294114	+	0.955770i	$0.595024\pi$
$702$	0	0
$703$	4008.66	0.215063
$704$	0	0
$705$	2762.90	0.147599
$706$	0	0
$707$	11357.8	0.604178
$708$	0	0
$709$	−35699.1	−1.89098	−0.945492	−	0.325645i	$-0.894419\pi$
$709$	−35699.1	−1.89098	−0.945492	+	0.325645i	$0.894419\pi$
$710$	0	0
$711$	−1215.53	−0.0641152
$712$	0	0
$713$	17984.4	0.944632
$714$	0	0
$715$	8677.56	0.453877
$716$	0	0
$717$	1658.20	0.0863691
$718$	0	0
$719$	−20861.4	−1.08206	−0.541028	−	0.841004i	$-0.681965\pi$
$719$	−20861.4	−1.08206	−0.541028	+	0.841004i	$0.681965\pi$
$720$	0	0
$721$	7577.48	0.391401
$722$	0	0
$723$	−972.001	−0.0499988
$724$	0	0
$725$	8719.25	0.446655
$726$	0	0
$727$	5381.54	0.274540	0.137270	−	0.990534i	$-0.456167\pi$
$727$	5381.54	0.274540	0.137270	+	0.990534i	$0.456167\pi$
$728$	0	0
$729$	−19362.6	−0.983724
$730$	0	0
$731$	6250.24	0.316243
$732$	0	0
$733$	−9737.69	−0.490682	−0.245341	−	0.969437i	$-0.578900\pi$
$733$	−9737.69	−0.490682	−0.245341	+	0.969437i	$0.578900\pi$
$734$	0	0
$735$	−1062.64	−0.0533282
$736$	0	0
$737$	14688.9	0.734156
$738$	0	0
$739$	−36183.5	−1.80112	−0.900562	−	0.434728i	$-0.856844\pi$
$739$	−36183.5	−1.80112	−0.900562	+	0.434728i	$0.856844\pi$
$740$	0	0
$741$	923.587	0.0457879
$742$	0	0
$743$	−1287.36	−0.0635649	−0.0317825	−	0.999495i	$-0.510118\pi$
$743$	−1287.36	−0.0635649	−0.0317825	+	0.999495i	$0.510118\pi$
$744$	0	0
$745$	−46532.3	−2.28834
$746$	0	0
$747$	20140.1	0.986465
$748$	0	0
$749$	10063.0	0.490913
$750$	0	0
$751$	−12722.8	−0.618193	−0.309096	−	0.951031i	$-0.600027\pi$
$751$	−12722.8	−0.618193	−0.309096	+	0.951031i	$0.600027\pi$
$752$	0	0
$753$	−1359.53	−0.0657954
$754$	0	0
$755$	−7104.48	−0.342461
$756$	0	0
$757$	−30073.5	−1.44391	−0.721955	−	0.691940i	$-0.756756\pi$
$757$	−30073.5	−1.44391	−0.721955	+	0.691940i	$0.756756\pi$
$758$	0	0
$759$	−310.814	−0.0148641
$760$	0	0
$761$	7169.59	0.341521	0.170760	−	0.985313i	$-0.445378\pi$
$761$	7169.59	0.341521	0.170760	+	0.985313i	$0.445378\pi$
$762$	0	0
$763$	15044.3	0.713812
$764$	0	0
$765$	−40479.1	−1.91311
$766$	0	0
$767$	−2502.71	−0.117820
$768$	0	0
$769$	−26826.5	−1.25798	−0.628991	−	0.777412i	$-0.716532\pi$
$769$	−26826.5	−1.25798	−0.628991	+	0.777412i	$0.716532\pi$
$770$	0	0
$771$	77.5128	0.00362070
$772$	0	0
$773$	4924.47	0.229134	0.114567	−	0.993415i	$-0.463452\pi$
$773$	4924.47	0.229134	0.114567	+	0.993415i	$0.463452\pi$
$774$	0	0
$775$	−60335.5	−2.79654
$776$	0	0
$777$	−122.127	−0.00563870
$778$	0	0
$779$	−55783.6	−2.56567
$780$	0	0
$781$	−12593.1	−0.576976
$782$	0	0
$783$	452.923	0.0206720
$784$	0	0
$785$	5452.66	0.247916
$786$	0	0
$787$	13223.3	0.598931	0.299465	−	0.954107i	$-0.403192\pi$
$787$	13223.3	0.598931	0.299465	+	0.954107i	$0.403192\pi$
$788$	0	0
$789$	1069.18	0.0482431
$790$	0	0
$791$	10417.4	0.468269
$792$	0	0
$793$	2800.02	0.125387
$794$	0	0
$795$	2221.24	0.0990932
$796$	0	0
$797$	−3635.34	−0.161569	−0.0807844	−	0.996732i	$-0.525743\pi$
$797$	−3635.34	−0.161569	−0.0807844	+	0.996732i	$0.525743\pi$
$798$	0	0
$799$	−37733.6	−1.67074
$800$	0	0
$801$	−19581.0	−0.863744
$802$	0	0
$803$	1827.06	0.0802934
$804$	0	0
$805$	−20587.3	−0.901375
$806$	0	0
$807$	−818.027	−0.0356827
$808$	0	0
$809$	35834.3	1.55731	0.778656	−	0.627451i	$-0.215902\pi$
$809$	35834.3	1.55731	0.778656	+	0.627451i	$0.215902\pi$
$810$	0	0
$811$	−1596.86	−0.0691408	−0.0345704	−	0.999402i	$-0.511006\pi$
$811$	−1596.86	−0.0691408	−0.0345704	+	0.999402i	$0.511006\pi$
$812$	0	0
$813$	1087.33	0.0469057
$814$	0	0
$815$	52482.2	2.25567
$816$	0	0
$817$	9079.33	0.388795
$818$	0	0
$819$	10321.2	0.440357
$820$	0	0
$821$	29012.8	1.23332	0.616660	−	0.787230i	$-0.288485\pi$
$821$	29012.8	1.23332	0.616660	+	0.787230i	$0.288485\pi$
$822$	0	0
$823$	20446.9	0.866018	0.433009	−	0.901390i	$-0.357452\pi$
$823$	20446.9	0.866018	0.433009	+	0.901390i	$0.357452\pi$
$824$	0	0
$825$	1042.74	0.0440043
$826$	0	0
$827$	27164.0	1.14218	0.571091	−	0.820887i	$-0.306520\pi$
$827$	27164.0	1.14218	0.571091	+	0.820887i	$0.306520\pi$
$828$	0	0
$829$	−23519.2	−0.985350	−0.492675	−	0.870213i	$-0.663981\pi$
$829$	−23519.2	−0.985350	−0.492675	+	0.870213i	$0.663981\pi$
$830$	0	0
$831$	−1129.52	−0.0471512
$832$	0	0
$833$	14512.8	0.603646
$834$	0	0
$835$	−20014.2	−0.829485
$836$	0	0
$837$	−3134.14	−0.129429
$838$	0	0
$839$	40135.2	1.65152	0.825758	−	0.564025i	$-0.190748\pi$
$839$	40135.2	1.65152	0.825758	+	0.564025i	$0.190748\pi$
$840$	0	0
$841$	−23428.0	−0.960599
$842$	0	0
$843$	−1841.62	−0.0752418
$844$	0	0
$845$	24329.2	0.990475
$846$	0	0
$847$	13934.2	0.565269
$848$	0	0
$849$	546.625	0.0220967
$850$	0	0
$851$	3102.08	0.124957
$852$	0	0
$853$	15717.0	0.630877	0.315439	−	0.948946i	$-0.397848\pi$
$853$	15717.0	0.630877	0.315439	+	0.948946i	$0.397848\pi$
$854$	0	0
$855$	−58801.5	−2.35201
$856$	0	0
$857$	16135.5	0.643149	0.321574	−	0.946884i	$-0.395788\pi$
$857$	16135.5	0.643149	0.321574	+	0.946884i	$0.395788\pi$
$858$	0	0
$859$	16972.7	0.674156	0.337078	−	0.941477i	$-0.390561\pi$
$859$	16972.7	0.674156	0.337078	+	0.941477i	$0.390561\pi$
$860$	0	0
$861$	1699.49	0.0672687
$862$	0	0
$863$	−26453.8	−1.04345	−0.521726	−	0.853113i	$-0.674712\pi$
$863$	−26453.8	−1.04345	−0.521726	+	0.853113i	$0.674712\pi$
$864$	0	0
$865$	−15932.2	−0.626255
$866$	0	0
$867$	−176.015	−0.00689480
$868$	0	0
$869$	−617.679	−0.0241120
$870$	0	0
$871$	−33777.0	−1.31399
$872$	0	0
$873$	−21890.7	−0.848669
$874$	0	0
$875$	38373.5	1.48258
$876$	0	0
$877$	41598.3	1.60168	0.800841	−	0.598877i	$-0.204386\pi$
$877$	41598.3	1.60168	0.800841	+	0.598877i	$0.204386\pi$
$878$	0	0
$879$	1682.08	0.0645452
$880$	0	0
$881$	20347.8	0.778131	0.389066	−	0.921210i	$-0.372798\pi$
$881$	20347.8	0.778131	0.389066	+	0.921210i	$0.372798\pi$
$882$	0	0
$883$	−16860.0	−0.642563	−0.321282	−	0.946984i	$-0.604114\pi$
$883$	−16860.0	−0.642563	−0.321282	+	0.946984i	$0.604114\pi$
$884$	0	0
$885$	−434.389	−0.0164993
$886$	0	0
$887$	32933.0	1.24665	0.623326	−	0.781962i	$-0.285781\pi$
$887$	32933.0	1.24665	0.623326	+	0.781962i	$0.285781\pi$
$888$	0	0
$889$	−29981.5	−1.13110
$890$	0	0
$891$	−9893.57	−0.371994
$892$	0	0
$893$	−54813.2	−2.05403
$894$	0	0
$895$	91076.0	3.40149
$896$	0	0
$897$	714.712	0.0266037
$898$	0	0
$899$	−6649.63	−0.246693
$900$	0	0
$901$	−30335.9	−1.12168
$902$	0	0
$903$	−276.608	−0.0101937
$904$	0	0
$905$	−43874.6	−1.61154
$906$	0	0
$907$	−11843.3	−0.433573	−0.216787	−	0.976219i	$-0.569558\pi$
$907$	−11843.3	−0.433573	−0.216787	+	0.976219i	$0.569558\pi$
$908$	0	0
$909$	25103.6	0.915990
$910$	0	0
$911$	29397.3	1.06913	0.534565	−	0.845127i	$-0.320476\pi$
$911$	29397.3	1.06913	0.534565	+	0.845127i	$0.320476\pi$
$912$	0	0
$913$	10234.3	0.370983
$914$	0	0
$915$	485.993	0.0175589
$916$	0	0
$917$	−15522.9	−0.559010
$918$	0	0
$919$	−9921.61	−0.356130	−0.178065	−	0.984019i	$-0.556984\pi$
$919$	−9921.61	−0.356130	−0.178065	+	0.984019i	$0.556984\pi$
$920$	0	0
$921$	−224.354	−0.00802683
$922$	0	0
$923$	28957.8	1.03267
$924$	0	0
$925$	−10407.1	−0.369928
$926$	0	0
$927$	16748.2	0.593400
$928$	0	0
$929$	42574.7	1.50359	0.751793	−	0.659399i	$-0.229189\pi$
$929$	42574.7	1.50359	0.751793	+	0.659399i	$0.229189\pi$
$930$	0	0
$931$	21081.8	0.742135
$932$	0	0
$933$	1976.69	0.0693612
$934$	0	0
$935$	−20569.7	−0.719468
$936$	0	0
$937$	−4500.02	−0.156893	−0.0784467	−	0.996918i	$-0.524996\pi$
$937$	−4500.02	−0.156893	−0.0784467	+	0.996918i	$0.524996\pi$
$938$	0	0
$939$	−1760.43	−0.0611816
$940$	0	0
$941$	2809.13	0.0973168	0.0486584	−	0.998815i	$-0.484505\pi$
$941$	2809.13	0.0973168	0.0486584	+	0.998815i	$0.484505\pi$
$942$	0	0
$943$	−43167.8	−1.49071
$944$	0	0
$945$	3587.74	0.123502
$946$	0	0
$947$	−5436.73	−0.186558	−0.0932789	−	0.995640i	$-0.529735\pi$
$947$	−5436.73	−0.186558	−0.0932789	+	0.995640i	$0.529735\pi$
$948$	0	0
$949$	−4201.30	−0.143709
$950$	0	0
$951$	2584.68	0.0881326
$952$	0	0
$953$	−52997.6	−1.80143	−0.900714	−	0.434413i	$-0.856956\pi$
$953$	−52997.6	−1.80143	−0.900714	+	0.434413i	$0.856956\pi$
$954$	0	0
$955$	−24288.1	−0.822979
$956$	0	0
$957$	114.921	0.00388179
$958$	0	0
$959$	212.048	0.00714015
$960$	0	0
$961$	16223.1	0.544564
$962$	0	0
$963$	22241.8	0.744270
$964$	0	0
$965$	11696.4	0.390176
$966$	0	0
$967$	35540.3	1.18190	0.590951	−	0.806707i	$-0.298753\pi$
$967$	35540.3	1.18190	0.590951	+	0.806707i	$0.298753\pi$
$968$	0	0
$969$	−2189.32	−0.0725810
$970$	0	0
$971$	25729.2	0.850351	0.425175	−	0.905111i	$-0.360212\pi$
$971$	25729.2	0.850351	0.425175	+	0.905111i	$0.360212\pi$
$972$	0	0
$973$	12789.8	0.421398
$974$	0	0
$975$	−2397.77	−0.0787590
$976$	0	0
$977$	−50377.3	−1.64965	−0.824827	−	0.565385i	$-0.808727\pi$
$977$	−50377.3	−1.64965	−0.824827	+	0.565385i	$0.808727\pi$
$978$	0	0
$979$	−9950.19	−0.324831
$980$	0	0
$981$	33251.6	1.08221
$982$	0	0
$983$	−30090.1	−0.976322	−0.488161	−	0.872754i	$-0.662332\pi$
$983$	−30090.1	−0.976322	−0.488161	+	0.872754i	$0.662332\pi$
$984$	0	0
$985$	−69876.6	−2.26036
$986$	0	0
$987$	1669.92	0.0538543
$988$	0	0
$989$	7025.99	0.225898
$990$	0	0
$991$	−28340.4	−0.908439	−0.454220	−	0.890890i	$-0.650082\pi$
$991$	−28340.4	−0.908439	−0.454220	+	0.890890i	$0.650082\pi$
$992$	0	0
$993$	2357.88	0.0753525
$994$	0	0
$995$	27825.6	0.886565
$996$	0	0
$997$	−53802.5	−1.70907	−0.854535	−	0.519394i	$-0.826158\pi$
$997$	−53802.5	−1.70907	−0.854535	+	0.519394i	$0.826158\pi$
$998$	0	0
$999$	−540.598	−0.0171209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	592.4.a.k.1.5		8
4.3	odd	2		296.4.a.d.1.4	✓	8
8.3	odd	2		2368.4.a.u.1.5		8
8.5	even	2		2368.4.a.x.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.4.a.d.1.4	✓	8	4.3	odd	2
592.4.a.k.1.5		8	1.1	even	1	trivial
2368.4.a.u.1.5		8	8.3	odd	2
2368.4.a.x.1.4		8	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 592 = 2^{4} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	592.a (trivial)

\(f(q)\)	\(=\)	\(q-0.270938 q^{3} -20.1562 q^{5} -12.1826 q^{7} -26.9266 q^{9} +O(q^{10})\)	\(q-0.270938 q^{3} -20.1562 q^{5} -12.1826 q^{7} -26.9266 q^{9} -13.6829 q^{11} +31.4637 q^{13} +5.46108 q^{15} -74.5832 q^{17} -108.342 q^{19} +3.30072 q^{21} -83.8400 q^{23} +281.273 q^{25} +14.6108 q^{27} +30.9993 q^{29} -214.509 q^{31} +3.70722 q^{33} +245.554 q^{35} -37.0000 q^{37} -8.52471 q^{39} +514.883 q^{41} -83.8023 q^{43} +542.738 q^{45} +505.926 q^{47} -194.585 q^{49} +20.2074 q^{51} +406.739 q^{53} +275.796 q^{55} +29.3540 q^{57} -79.5428 q^{59} +88.9920 q^{61} +328.035 q^{63} -634.189 q^{65} -1073.52 q^{67} +22.7154 q^{69} +920.355 q^{71} -133.528 q^{73} -76.2074 q^{75} +166.693 q^{77} +45.1424 q^{79} +723.059 q^{81} -747.964 q^{83} +1503.31 q^{85} -8.39888 q^{87} +727.198 q^{89} -383.309 q^{91} +58.1186 q^{93} +2183.77 q^{95} +812.977 q^{97} +368.435 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9}+O(q^{10}) \) 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9	\( 8 q - 7 q^{3} + 18 q^{5} + 23 q^{7} + 137 q^{9} - 33 q^{11} + 50 q^{13} - 22 q^{15} + 78 q^{17} - 158 q^{19} + 373 q^{21} - 76 q^{23} + 678 q^{25} - 469 q^{27} + 358 q^{29} - 390 q^{31} + 333 q^{33} - 546 q^{35} - 296 q^{37} - 442 q^{39} + 1141 q^{41} + 4 q^{43} + 1770 q^{45} + 735 q^{47} + 731 q^{49} - 132 q^{51} + 1349 q^{53} + 882 q^{55} + 2648 q^{57} - 1146 q^{59} + 1920 q^{61} + 534 q^{63} + 1748 q^{65} - 184 q^{67} + 1024 q^{69} + 881 q^{71} + 985 q^{73} + 1803 q^{75} + 1903 q^{77} + 658 q^{79} + 2296 q^{81} + 2773 q^{83} - 592 q^{85} + 3730 q^{87} + 234 q^{89} + 638 q^{91} - 904 q^{93} + 3660 q^{95} + 2174 q^{97} + 3582 q^{99}+O(q^{100}) \) 8 * q - 7 * q^3 + 18 * q^5 + 23 * q^7 + 137 * q^9 - 33 * q^11 + 50 * q^13 - 22 * q^15 + 78 * q^17 - 158 * q^19 + 373 * q^21 - 76 * q^23 + 678 * q^25 - 469 * q^27 + 358 * q^29 - 390 * q^31 + 333 * q^33 - 546 * q^35 - 296 * q^37 - 442 * q^39 + 1141 * q^41 + 4 * q^43 + 1770 * q^45 + 735 * q^47 + 731 * q^49 - 132 * q^51 + 1349 * q^53 + 882 * q^55 + 2648 * q^57 - 1146 * q^59 + 1920 * q^61 + 534 * q^63 + 1748 * q^65 - 184 * q^67 + 1024 * q^69 + 881 * q^71 + 985 * q^73 + 1803 * q^75 + 1903 * q^77 + 658 * q^79 + 2296 * q^81 + 2773 * q^83 - 592 * q^85 + 3730 * q^87 + 234 * q^89 + 638 * q^91 - 904 * q^93 + 3660 * q^95 + 2174 * q^97 + 3582 * q^99

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